Question

Prove:

A nonempty subset C⊆**R** is closed if and only if
there is a continuous function g:R→R such that
C=g^{-1}(0).

Answer #1

let A be a nonempty subset of R that is bounded below. Prove
that inf A = -sup{-a: a in A}

Let A be open and nonempty and f : A → R. Prove that f is
continuous at a if and only if f is both upper and lower
semicontinuous at a.

Prove that if f: X → Y is a continuous function and C ⊂ Y is
closed that the preimage of C, f^-1(C), is closed in X.

Prove or provide a counterexample
If A is a nonempty countable set, then A is closed in T_H.

let F : R to R be a continuous function
a) prove that the set {x in R:, f(x)>4} is open
b) prove the set {f(x), 1<x<=5} is connected
c) give an example of a function F that {x in r, f(x)>4} is
disconnected

a)Suppose U is a nonempty subset of the vector space V over
field F. Prove that U is a subspace if and only if cv + w ∈ U for
any c ∈ F and any v, w ∈ U
b)Give an example to show that the union of two subspaces of V
is not necessarily a subspace.

if f: D - R be continuous, and D is close, then F(D) is closed.
prove or give counterexample

Problem 6. For a closed convex nonempty subset
K of a Hilbert space H and x ∈ H, denote by P x ∈ K a unique
closest point to x among points in K, i.e. P x ∈ K such that
||P x − x|| ≤ ||y − x||, for all y ∈ K.
First show that such point P x exists and unique. Next prove
that all x, y ∈ H
||P x − P y|| ≤ ||x −...

Let f, g : X −→ C denote continuous functions from the open
subset X of C. Use the properties of limits given in section 16 to
verify the following:
(a) The sum f+g is a continuous function. (b) The product fg is
a continuous function.
(c) The quotient f/g is a continuous function, provided g(z) !=
0 holds for all z ∈ X.

We know that any continuous function f : [a, b] → R is uniformly
continuous on the finite closed interval [a, b]. (i) What is the
definition of f being uniformly continuous on its domain? (This
definition is meaningful for functions f : J → R defined on any
interval J ⊂ R.) (ii) Given a differentiable function f : R → R,
prove that if the derivative f ′ is a bounded function on R, then f
is uniformly...

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